A250001 -id:A250001 - cp's OEIS frontend

A261070 Irregular triangle read by rows: T(n,k) is the number of arrangements of n circles with 2k intersections (using the same rules as A250001).

1, 1, 2, 1, 4, 4, 2, 4, 9, 15, 15, 31, 24, 35, 44, 20, 50

Offset: 0

Benoit Jubin, Aug 08 2015

Length of n-th row: 1 + (n-1)n/2 (for a configuration for T(n,(n-1)n/2), consider n circles of radius 1 and centers at (k/n,0) for 1<=k<=n).

The generating function down the column k=1 is 1+z^2 *C^2(z) *[C^2(z)+C(z^2)]/ (2*[1-z*C(z)]) = 1+ z^2 +4*z^3 +15*z^4+ 50*z^5+...where C(z) = 1+z+2*z^2+4*z^3+... is the g.f. of A000081 divided by z; eq. (78) in arXiv:1603.00077. - R. J. Mathar, Mar 05 2016

			n\k 0  1  2  3  4  5  6
0   1
1   1
2   2  1
3   4  4  2  4
4   9 15 15 31 24 35 44
5  20 50  .  .  .  .  .  .  .  .  .

R. J. Mathar, Topologically Distinct Sets of Non-intersecting Circles in the Plane, arXiv:1603.00077 [math.CO], 2016.

Row sums give A250001.

Cf. A000081, A152947, A249752, A252158, A280786 (column k=1)

A250001(n) = Sum_{k>=0} T(n,k).

A000081(n+1) = T(n,0).

T(4,2)..T(5,0) (6 terms) from Travis Vasquez, Nov 28 2024

A260995 Erroneous version of A250001.

Original entry on oeis.org

1, 1, 3, 14, 168

Offset: 1

dead

Included in accordance with OEIS policy of including published but erroneous sequences, to serve as pointers to the correct versions.

A275923 Number of connected arrangements of n circles in the affine plane, in the sense that the union of the solid circles is a connected set.

Original entry on oeis.org

1, 1, 2, 11, 156, 16756

Offset: 0

N. J. A. Sloane, Aug 29 2016, based on information supplied by Jon Wild

Arrangements in A250001 that are connected (in the sense that the union of the solid circles is a connected set). See A250001, the main entry for this problem, for further information.

Is this also 1 together with column 1 of A285996? - Omar E. Pol, May 28 2017

Jon Wild, Figure showing relationship between A250001, A275923, A275924, and A288554 for n=3 circles

Cf. A250001, A275924, A288554-A288568.

a(5) from Jon Wild, Aug 31 2016

A275924 Number of connected arrangements of n circles in the affine plane, in the sense that the union of the boundaries of the circles is a connected set.

Original entry on oeis.org

1, 1, 1, 6, 112, 15502

Offset: 0

N. J. A. Sloane, Aug 29 2016, based on information supplied by Jon Wild

Arrangements in A250001 that are connected, in the sense that the union of the (boundaries of the) circles is a connected set. See A250001, the main entry for this problem, for further information.

Jon Wild, Figure showing relationship between A250001, A275923, A275924, and A288554 for n=3 circles

Cf. A250001, A275923, A288554-A288568.

a(5) corrected by Jon Wild, Aug 31 2016

A288554 Number of arrangements of n circles in the affine plane with the property that every circle intersects all the other circles.

Original entry on oeis.org

1, 1, 1, 4, 45, 5102

Offset: 0

N. J. A. Sloane, Jun 12 2017, based on information supplied by Jon Wild on Aug 31 2016

See A250001, the main entry for this problem, for further information.

Jon Wild, Figure showing relationship between A250001, A275923, A275924, and A288554 for n=3 circles

Cf. A250001, A275923, A275924, A288554-A288568.

A288568 Number of non-isomorphic connected arrangements of n pseudo-circles on a sphere, in the sense that the union of the pseudo-circles is a connected set, reduced for mirror symmetry.

Original entry on oeis.org

1, 1, 1, 3, 21, 984, 609423

Offset: 0

N. J. A. Sloane, Jun 13 2017, based on information supplied by Jon Wild on August 31 2016

These counts have been reduced for mirror symmetry. Computed up to n=5 by Jon Wild and Christopher Jones and communicated to N. J. A. Sloane on August 31 2016. Definition corrected Dec 10 2017 thanks to Manfred Scheucher, who has computed same result with Stefan Felsner independently.
The list of arrangements is available online on the Homepage of Pseudocircles (see below) and a detailed description for the enumeration can be found in Arrangements of Pseudocircles: On Circularizability (see below). - Manfred Scheucher, Dec 11 2017
See A250001, the main entry for this problem, for further information.

S. Felsner and M. Scheucher Homepage of Pseudocircles
S. Felsner and M. Scheucher, Arrangements of Pseudocircles: On Circularizability, arXiv:1712.02149 [cs.CG], 2017.

Cf. A250001, A275923, A275924, A288554-A288568, A296406, A296407-A296412, A006248.

a(n) = 2^(\Theta(n^2)). (cf. Arrangements of Pseudocircles: On Circularizability)

a(6) from Manfred Scheucher, Dec 11 2017

A296407 Number of digon-free connected arrangements of n pseudo-circles on a sphere, in the sense that the union of the pseudo-circles is a connected set, reduced for mirror symmetry.

Original entry on oeis.org

1, 1, 1, 1, 3, 30, 4509

Offset: 0

Manfred Scheucher, Dec 11 2017

For more information, see A288568.

Cf. A250001, A275923, A275924, A288554-A288568, A296406, A296407-A296412, A006248.

A296406 Number of non-isomorphic arrangements of n pairwise intersecting pseudo-circles on a sphere, reduced for mirror symmetry.

Original entry on oeis.org

1, 1, 1, 2, 8, 278, 145058, 447905202

Offset: 0

Manfred Scheucher, Dec 11 2017

The list of arrangements is available online on the Homepage of Pseudocircles (see below) and a detailed description for the enumeration can be found in Arrangements of Pseudocircles: On Circularizability (see below).

S. Felsner and M. Scheucher Homepage of Pseudocircles
S. Felsner and M. Scheucher, Arrangements of Pseudocircles: On Circularizability, arXiv:1712.02149 [cs.CG], 2017.
Yan Alves Radtke, Stefan Felsner, Johannes Obenaus, Sandro Roch, Manfred Scheucher, and Birgit Vogtenhuber, Flip Graph Connectivity for Arrangements of Pseudolines and Pseudocircles, arXiv:2310.19711 [math.CO], 2023. See p. 41.

Cf. A250001, A275923, A275924, A288554-A288568, A296407-A296412, A006248.

a(n) = 2^(\Theta(n^2)). (cf. Arrangements of Pseudocircles: On Circularizability)

A296412 Number of non-isomorphic digon-free cylindrical arrangements of n pairwise intersecting pseudo-circles on a sphere, in the sense that two cells of the arrangement are separated by each of the pseudo-circles, reduced for mirror symmetry.

Original entry on oeis.org

1, 1, 1, 1, 2, 14, 2131, 3012906

Offset: 0

Manfred Scheucher, Dec 11 2017

For more information, see A296406.

Cf. A250001, A275923, A275924, A288554-A288568, A296406, A296407-A296412, A006248.

A048872 Number of non-isomorphic arrangements of n lines in the real projective plane such that the lines do not all pass through a common point.

Original entry on oeis.org

1, 2, 4, 17, 143, 4890, 460779

Offset: 3

N. J. A. Sloane

J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, CRC Press, 1997, p. 102.
B. Grünbaum, Arrangements and Spreads. American Mathematical Society, Providence, RI, 1972, p. 4.

Stefan Felsner and Jacob E. Goodman, Pseudoline Arrangements, Chapter 5 of Handbook of Discrete and Computational Geometry, CRC Press, 2017, see Table 5.6.1. [Specific reference for this sequence] - _N. J. A. Sloane_, Nov 14 2023
Jacob E. Goodman, Joseph O'Rourke, and Csaba D. Tóth, editors, Handbook of Discrete and Computational Geometry, CRC Press, 2017, see Table 5.6.1. [General reference for 2017 edition of the Handbook]
N. J. A. Sloane, Illustration of a(3) - a(6) [based on Fig. 2.1 of Grünbaum, 1972]

See A132346 for the sequence when we include the arrangement where the lines do pass through a common point, which is 1 greater than this.

Cf. A003036, A048873, A090338, A090339, A241600, A250001, A018242, A063800 (arrangements of pseudolines).

a(7)-a(9) from Handbook of Discrete and Computational Geometry, 2017, by Andrey Zabolotskiy, Oct 09 2017

cp's OEIS Frontend

A261070 Irregular triangle read by rows: T(n,k) is the number of arrangements of n circles with 2k intersections (using the same rules as A250001).

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A260995 Erroneous version of A250001.

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A275923 Number of connected arrangements of n circles in the affine plane, in the sense that the union of the solid circles is a connected set.

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A275924 Number of connected arrangements of n circles in the affine plane, in the sense that the union of the boundaries of the circles is a connected set.

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A288554 Number of arrangements of n circles in the affine plane with the property that every circle intersects all the other circles.

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A288568 Number of non-isomorphic connected arrangements of n pseudo-circles on a sphere, in the sense that the union of the pseudo-circles is a connected set, reduced for mirror symmetry.

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A296407 Number of digon-free connected arrangements of n pseudo-circles on a sphere, in the sense that the union of the pseudo-circles is a connected set, reduced for mirror symmetry.

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A296406 Number of non-isomorphic arrangements of n pairwise intersecting pseudo-circles on a sphere, reduced for mirror symmetry.

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A296412 Number of non-isomorphic digon-free cylindrical arrangements of n pairwise intersecting pseudo-circles on a sphere, in the sense that two cells of the arrangement are separated by each of the pseudo-circles, reduced for mirror symmetry.

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A048872 Number of non-isomorphic arrangements of n lines in the real projective plane such that the lines do not all pass through a common point.

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